Question

If $L=\underset{x\to 3^{-}}{lim}\frac{exp((2x-2)\ln 3{)}^{\frac{\left[x\right]}{4}}-9}{3^x-27},$
where [.] denotes the greatest integer function then the value of $9L$ is

Answer 1

$L=\underset{x\to 3^{-}}{lim}\frac{exp((2x-2)\ln 3{)}^{\frac{\left[x\right]}{4}}-9}{3^x-27},$
$\Rightarrow L=\underset{x\to 3^{-}}{lim}\frac{exp(2(x-1)\ln 3{)}^{\frac{2}{4}}-9}{3^x-27}\Rightarrow L=\underset{x\to 3^{-}}{lim}\frac{(3^{2x-2}{)}^{\frac{1}{2}}-9}{3^x-27}\Rightarrow L=\underset{x\to 3^{-}}{lim}\frac{3^{x-1}-9}{3^x-27}$

Using L'Hospital's rule ,
$\Rightarrow L=\underset{x\to 3^{-}}{lim}\frac{3^{x-1}\ln 3}{3^x\ln 3}\Rightarrow L=\underset{x\to 3^{-}}{lim}\frac{1}{3}$
$\therefore 9L=3$